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720 CHAPTER 21 Series Representations of Functions

In some cases, we have extended important real-valued functions to the complex plane.
x
z
Examples are e ;sin(z) and cos(z), which agree with e ; and sin(x) and cos(x), respectively,
when z = x is real. In such cases, we can obtain the power series expansion of the complex
function directly from the expansion for the real-valued function, if this is known. For example,
using the familiar Maclaurin series, we can immediately write
∞
1
z n
e = z ,
n!
n=0
∞ n
(−1)

sin(z) = z 2n+1 ,
(2n + 1)!
n=0
and
∞ n
(−1)

2n
cos(z) = z .
(2n)!
n=0
n
We rarely compute Taylor coefficients as f (z 0 )/n! in expanding f (z) about z 0 . If possible,
we use known series with algebra and calculus operations. Of course, term by term differentiation
and integration of power series applies to Taylor series.
EXAMPLE 21.2
∞ 1
z 2 2n
e = z
n!
n=0
by replacing z with z in the Maclaurin expansion of e .
z
2
EXAMPLE 21.3
Start with the familiar geometric series
∞
1
3
n
2
= z = 1 + z + z + z + ···
1 − z
n=0
for |z| < 1. Differentiate to obtain
∞
1
2
= nz n=1 = 1 + 2z + 3z + ···
(1 − z) 2
n=1
for |z| < 1. Differentiate again to obtain
∞
2
2
= n(n − 1)z n−2 = 2 + 6z + 12z + ···
(1 − z) 3
n=2
for |z| < 1.
Replacing z with −z, we obtain
∞
1 n n
= (−1) z
1 + z
n=0
for |z| < 1. This is also called a geometric series. Now differentiation yields
∞
−1 n n−1
= n(−1) z
(1 + z) 2
n=1
for |z| < 1.
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October 14, 2010 15:35 THM/NEIL Page-720 27410_21_ch21_p715-728